Problem 41 - Fundamentals of Mathematics

Question

Determine the value of each of the powers. Use a calculator to check each result. \(10^{3}\)

Step-by-Step Solution

Answer

The value of \(10^3\) is 1000.

Step 1: Understand the Power

The given problem is to determine the value of the exponential expression \(10^3\). This expression consists of a base number (10) and an exponent (3).

Step 2: Apply the Exponent

The exponent 3 indicates how many times the base (10) is multiplied by itself. Mathematically, this means \(10\) is used as a factor three times: \(10 \times 10 \times 10\).

Step 3: Calculate the Power Manually

Perform the multiplication: First, multiply \(10\) by \(10\), which gives \(100\). Then multiply \(100\) by \(10\), resulting in \(1000\). Therefore, \(10^3 = 1000\).

Step 4: Verify with a Calculator

Use a calculator to confirm your manual calculation. Enter \(10\), then raise it to the power of \(3\) using the calculator's exponent function. The calculator should display \(1000\), verifying that \(10^3 = 1000\).

Key Concepts

Understanding Base and Exponent

In any expression involving exponents, you will encounter two fundamental components: the base and the exponent. These elements are crucial to understanding how exponents work.
'Base' refers to the number that you are multiplying. It's your starting point. In our example, this is the number 10.
'Exponent' refers to the power to which the base number is raised. It indicates how many times you multiply the base by itself. Here, it's the number 3.
To express this in mathematical terms, when we write it as \(10^3\), it means we multiply 10 by itself three times. This concept lets us write large multiplications concisely. For instance, instead of writing out 10 multiplied several times, we use this notation to simplify and streamline computation.

Multiplication in Exponents

Once you understand what each part of the expression \(10^3\) stands for, the next step is to apply that knowledge by performing multiplication.
Starting with the base (which is 10 in our scenario), you multiply it by itself as many times as the exponent dictates. So, \(10^3\) requires multiplying 10 by itself three times:

• First, calculate \(10 \times 10\), which equals 100.
• Next, multiply that result (100) by 10, giving you 1000.

This sequence of multiplications translates the expression into an easily understandable calculation.
Understanding this multiplication process is key to solving any problems involving exponents, taking complex equations and breaking them down into a series of simpler steps.

Using a Calculator for Verification

Once you've manually calculated an exponent, it's wise to verify your results with a calculator to ensure complete accuracy.
Most scientific calculators have an exponent function that makes this process quick and error-free. To check your answer for \(10^3\):

• Begin by entering the base, which is 10.
• Next, utilize the exponent button, often labeled as \(^\), \(\text{EXP}\), or something similar.
• Input the exponent, which is 3.
• Press 'Enter' or '=' to complete the calculation.
• The calculator should display 1000, verifying your manual calculation was correct.

This verification step serves as a comforting confirmation, ensuring that your understanding and manual work align with the technological accuracy of a calculator.